When Marty asked the question about a definition of grouped coloring I looked, and was surprised to find very little. Here is my effort to write something down. This is a first draft. As always, comments and further contributions are welcome. Keith

If a candidate occurs only twice in a row, column, or box, the two cells in which it occurs are said to be a "strong link" in that candidate. Strong links are a fundamental building block of many Sudoku techniques. An excellent guide has been written by Havard: Strong Links for Beginners

Code:

Notation: In the following,
. Cell may be anything
* Cell contains the candidate
/ Cell does not contain the candidate
@ Candidate is eliminated from this cell

The logic is: The candidate in B1 must lie in either R1 or C1. Either way, one of the candidates in B3 or B7 is true, and @ can be eliminated.

This is "grouped" coloring. The candidates in B1 can be grouped into those in R1 or those in C1. (in the last sentence, "or" rather than "and" is very important. The groups do not share candidates.) For the purposes of this chain, the cells in R1B1 and C1B1 act like a strong link.

It is important to note that "grouped" links only apply in the context of a particular chain.

In our coloring diagram, we can add two more strong links in R2 and C2, making an elimination in the cell labelled #:

Simple coloring (grouped or not) no longer eliminates #. The groups for the elimination of @ are not relevant for the elimination of #.

In diagram 4, # is eliminated by what I would call strong links/kite/multi-coloring/turbot fish or whatever other terms are used.

In diagram 5, the cells that made up the group are added and you say that simple coloring (grouped or not) no longer eliminates #. That I don't understand since it looks like the same logic eliminates # in both diagrams 4 and 5.

Diagram 4 does not require multi-coloring for the # elimination; it is accomplished by "basic" (single cluster) coloring, which is what I believe Keith meant by simple coloring. All of the links connecting the "pincers" are conjugate. Diagram 5 requires ("non-simple") multi-coloring due to the weak (only) link in Box 1.

Regarding the ER, I will attempt to provide additional explanation according to my understanding of such things:

Yes, the ER, too, uses groups. To understand how requires a bit of preliminary clarification of definitions. The strong links discussed by Keith are conjugate links, meaning that one side is true and the other false. It is true that conjugate links are strong links. However, conjugate links are also weak links. Conjugate links satisfy the logical requirements of both strong and weak links (which is why they are so easy to use). [See below if you need more explanation of strong and weak inferences.]

There also exist strong links that are only strong links. The ER shown above in Diagram 6 is an example of one. But first, let's look at the ER structure in Box 1 of Diagram 5. It can be expressed as a grouped conjugate link between r1c23 on the one side and r23c1 on the other. As such, it is both a grouped strong link and a grouped weak link. (When it functions as an ER, it is functioning as a grouped strong link. For the elimination at @, it is functioning as a grouped weak link, not as an ER.)

[To be precise, the grouped link is between the grouped candidate digit in cells r1c23 and the grouped candidate digit in cells r23c1. Saying that the link is between the grouped cells is a shorthand for this.]

Since Keith was considering only weak links and conjugate strong links, his statement that the groups on either side of a link cannot share a candidate is true. To understand the grouped link of the ER in Diagram 6, we must consider the case where this restriction does not apply. Here, the grouped link is between r1c123 and r123c1. Notice that both sides of the link contain the candidate in r1c1. Thus, if r1c1 is true, then both sides of the link are true. Still, since this link includes all instances of the candidate in Box 1, it is impossible for both sides to be false. This is a "strong only" link, sometimes called a "strong inference" link. But, really "strong link" is a perfectly fine description provided it is understood that strong link is not a synonym for conjugate link.
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True and False as applied to groups:

It might help to explain what it means for a group to be true or false. A group is true if at least one member of the group is true. A group is false if all members of the group are false.
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Strong and Weak Inferences:

For those not familiar with strong and weak inference concepts, this might be a good place to provide a quickie explanation. A strong inference exists between two things if when one of those things is assumed to be false the other must be true. The minimum requirement for this occurs when it is impossible for both things to be false. A conjugate link, such as exists between two candidates in a House which are the only two instances of that candidate in that House, meets this requirement: if they were both false, then the House would lack that candidate, which is not possible. A grouped link that encompases all of the instances of a candidate in a House does the same. Note that a link in which both sides can be true also satisfies this requirement as long as it is impossible for both sides to be false.

A weak inference exists between two things if when one of those things is assumed to be true the other must be false. The minimum requirement for this occurs when it is impossible for both things to be true. A conjugate link satisfies this requirement and is thus also a weak link. Any two candidates within a House that contains more than two of those candidates is also a weak link; both sides might be false, but both sides cannot be true. And, a grouped link within a House that does not share a candidate on both sides of the link is also a weak link, whether or not it includes all instances of the candidate digit within that House.

The examples given for strong and weak links do not exhaust the possibilities. But, more some other time!

Diagram 4 does not require multi-coloring for the # elimination; it is accomplished by "basic" (single cluster) coloring, which is what I believe Keith meant by simple coloring. All of the links connecting the "pincers" are conjugate. Diagram 5 requires ("non-simple") multi-coloring due to the weak (only) link in Box 1.

Thanks, I just missed that chain in diagram 4. But I deserve a dozen lashes with a wet noodle because the fact that there were just two possibilities in box 1 meant there was a simple chain.

I deliberately avoided using the full Empty Rectangle pattern in another post on grouped strong links because I knew that it would be difficult to explain. However, since Pandora's Box (pun intended) has been opened here, I would like to add my opinion.

The two ER patterns in this loop can not be included in an AIC without some confusion.

This is true only if you want to insist that "strong link" is synonymous with "conjugate link". And, if that is your opinion, then fine. I was careful to explain that I do not adhere to that restriction. In my opinion, this limitation of strong link to conjugate links is an understandable accident of sudoku history: conjugate links are naturally the first sort of strong links to have gained attention. But, they are called strong because they can be used for strong inferences, which the also-initially-obvious weak (only) links cannot be.

The statement that "a strong link can be used as both a weak and strong inference" is, to me, a syntactical muddle. However, the statement "a conjugate link can be used as both a weak and strong inference" is not only coherent, it is informative and makes things clearer. There is a universe of strong links and a universe of weak links; the overlap of these two universes is the universe of conjugate links.

If one defines strong link (as I do ... and I am certainly not the only one) as a link with a strong inference, then there is no need to worry about including the ERI digit on both sides of the grouped link and the two consecutive ERs can be written in an AIC without any problem whatsoever.

Similar strong (only) links are used in AICs all the time. For instance, the strong links induced by Deadly Patterns are such strong links: it is possible that both sides of the link are true. Are all the AICs that include such DP-induced strong links also "problematic"?

[Note: I should point out that this entire discussion is concerned only with bidirectional links.]

This is multicoloring; it is also a kite / skyscraper, etc. The logic is: Either A and/or B is false. Therefore, given the strong links, either a and/or b is true. So, # is eliminated.

You are correct: The same multicoloring logic can be applied in both diagrams, to make the same elimination. But, suppose you wanted to use the pincer cells in some other pattern. The logic following simple coloring (One of a and A is true, the other is false, Diagram 4a) is (subtly) different than the logic following multicoloring (One, or both, of a and b are true, Diagram 5a).